{
  "id": "2010.09910",
  "version": "v1",
  "published": "2020-10-19T22:56:33.000Z",
  "updated": "2020-10-19T22:56:33.000Z",
  "title": "Families of polynomials of every degree with no rational preperiodic points",
  "authors": [
    "Mohammad Sadek"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be a number field. Given a polynomial $f(x)\\in K[x]$ of degree $d\\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\\mathbb Q]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\\mathbb Q$. In this article, given any integer $d\\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\\in K(t)$, with no rational preperiodic points for any $t\\in K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-19T22:56:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational preperiodic points",
      "polynomial",
      "parametric families",
      "number field",
      "uniform bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}